- Identify the quadratic relation \(y = -0.1x^2 + 10\) as a downward-opening parabola.
- Calculate the peak height by evaluating the vertex at \(x = 0\), resulting in \(10 \text{ m}\).
- Determine the width by solving for the x-intercepts \(x = \pm 10\), yielding a total span of \(\boxed{20 \text{ m}}\).
- The arch has a maximum height of \(\boxed{10 \text{ m}}\) and a total width of \(\boxed{20 \text{ m}}\).
Explanation
- Analysis of the Bridge Arch Quadratic Function
The arch of the bridge is described by the quadratic function \(y = -0.1x^2 + 10\), where \(y\) is the height in meters and \(x\) is the horizontal distance from the center in meters. This is a quadratic equation of the form \(y = ax^2 + c\), where \(a = -0.1\) and \(c = 10\). Since \(a < 0\), the graph of this function is a downward-opening parabola. The vertex of this parabola represents the highest point, which occurs at the center of the arch (\(x = 0\)). Evaluating the function at \(x = 0\) gives \(y = -0.1(0)^2 + 10 = 10\), so the maximum height is 10 meters. To find where the arch meets the ground, we set \(y = 0\) and solve for \(x\): \(0 = -0.1x^2 + 10\), which simplifies to \(0.1x^2 = 10\), so \(x^2 = 100\), yielding roots \(x = 10\) and \(x = -10\). These are the points where the arch touches the ground.
- Describing the Shape of the Arch
The shape of the arch is a parabola that opens downwards. This is a common shape for structural arches, as it efficiently distributes weight and stress. The vertex at \((0, 10)\) is the peak of the bridge, and the curve slopes symmetrically down to the ground level on both sides. The symmetry around the y-axis confirms that the arch is balanced from the center.
- Calculating Height and Width
The total height of the arch is the value of \(y\) at the peak (vertex), which is 10 meters. The total width of the arch is determined by the distance between the two points where the arch meets the ground. We found these points to be \(x = 10\) and \(x = -10\). The width is the difference between these two horizontal positions: \(\text{Width} = |10 - (-10)| = 20\) meters. Thus, the arch is 10 meters high and 20 meters wide.
Examples
In structural engineering, parabolic arches are frequently used to design bridges because they can support significant loads by transforming compressive forces along the structure. If you are a civil engineer designing a bridge, understanding the quadratic equation governing its shape allows you to calculate how much material is needed for construction, determine clearance for traffic passing underneath, and ensure the structure remains stable. By mastering these calculations, you can predict the dimensions of various architectural components and optimize them for safety and efficiency.